What is the difference between "subscription" and "membership"?

A friend of mine is trying to coin the correct English word for the people paying for his company's services.

My initial choice was "subscriptions", but I'm not so sure about "membership".

Do you know what the difference between these two words is?

Subscription implies a service that is paid for on a regular basis. Membership may be paid for, but that is not implied.

I could be a member of the Zooey Deschanel fan club for free, but subscribe to the monthly Zooey Deschanel magazine for $20 a year, for example. A man can dream...

Also, one can be described as a member of organisations that actually pay you, like the armed forces or emergency services.

Members of a subscription system are generally referred to as subscribers, but this is context sensitive. Your friend could simply refer to them as customers, as that's the usual word for someone paying for goods or services.

The NOAD describes subscription as:

subscription /səbˈskrɪpʃən/
noun
1. the action of making or agreeing to make an advance payment in order to receive or participate in something: the newsletter is available only on subscription | take out a one-year subscription.
• an arrangement by which access is granted to an online service.
• chiefly Brit. a payment of such a type: membership is available at an annual subscription of £300.
• a system in which the production of a book is wholly or partly financed by advance orders.
2 formal a signature or short piece of writing at the end of a document: he signed the letter and added a subscription.
• archaic a signed declaration or agreement.

The definition of membership is the following:

membership /ˈmɛmbərˌʃɪp/
noun
the fact of being a member of a group: Taiwan has applied for membership in the World Trade Organization | a membership card.
• the number or body of members in a group: our membership has grown by 600,000 in the past 18 months.

The word you are looking for is probably subscribers.

Rather than using membership, you should use members, but that word doesn't implicates they pay for any service.